矩阵秩的基本性质 性质1：初等变换不改变矩阵的秩
性质2：矩阵的行秩等于矩阵的列秩等于矩阵的秩
定理 若 A A A B B B R ( A ) = R ( B ) R( A )=R( B ) R ( A ) = R ( B )
证明：已知矩阵的初等行变换不改变知阵的秩．对矩阵 A A A B B B A T A ^{ T } A T B T B ^{ T } B T R ( A ) = R ( A T ) , R ( B ) = R ( B T ) R( A )=R\left( A ^{ T }\right), R( B )=R\left( B ^{ T }\right) R ( A ) = R ( A T ) , R ( B ) = R ( B T ) A A A B B B R ( A ) = R ( B ) R( A )=R( B ) R ( A ) = R ( B )
因此，若 A ∼ B A \sim B A ∼ B R ( A ) = R ( B ) R( A )=R( B ) R ( A ) = R ( B )
例 求矩阵 A = ( 3 0 − 2 − 1 3 1 − 1 3 2 0 1 0 1 − 1 1 2 − 2 1 6 0 ) A=\left(\begin{array}{ccccc}3 & 0 & -2 & -1 & 3 \\ 1 & -1 & 3 & 2 & 0 \\ 1 & 0 & 1 & -1 & 1 \\ 2 & -2 & 1 & 6 & 0\end{array}\right) A = 3 1 1 2 0 − 1 0 − 2 − 2 3 1 1 − 1 2 − 1 6 3 0 1 0
解：
A = ( 3 0 − 2 − 1 3 1 − 1 3 2 0 1 0 1 − 1 1 2 − 2 1 6 0 ) → ( 1 0 1 − 1 1 1 − 1 3 2 0 3 0 − 2 − 1 3 2 − 2 1 6 0 ) → ( 1 0 1 − 1 1 0 − 1 2 3 − 1 0 0 − 5 2 0 0 − 2 − 1 8 − 2 ) → ( 1 0 1 − 1 1 0 − 1 2 3 − 1 0 0 − 5 2 0 0 0 0 0 0 ) , A =\left(\begin{array}{ccccc}
3 & 0 & -2 & -1 & 3 \\
1 & -1 & 3 & 2 & 0 \\
1 & 0 & 1 & -1 & 1 \\
2 & -2 & 1 & 6 & 0
\end{array}\right) \rightarrow\left(\begin{array}{ccccc}
1 & 0 & 1 & -1 & 1 \\
1 & -1 & 3 & 2 & 0 \\
3 & 0 & -2 & -1 & 3 \\
2 & -2 & 1 & 6 & 0
\end{array}\right) \rightarrow\left(\begin{array}{ccccc}
1 & 0 & 1 & -1 & 1 \\
0 & -1 & 2 & 3 & -1 \\
0 & 0 & -5 & 2 & 0 \\
0 & -2 & -1 & 8 & -2
\end{array}\right) \rightarrow\left(\begin{array}{ccccc}
1 & 0 & 1 & -1 & 1 \\
0 & -1 & 2 & 3 & -1 \\
0 & 0 & -5 & 2 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right), A = 3 1 1 2 0 − 1 0 − 2 − 2 3 1 1 − 1 2 − 1 6 3 0 1 0 → 1 1 3 2 0 − 1 0 − 2 1 3 − 2 1 − 1 2 − 1 6 1 0 3 0 → 1 0 0 0 0 − 1 0 − 2 1 2 − 5 − 1 − 1 3 2 8 1 − 1 0 − 2 → 1 0 0 0 0 − 1 0 0 1 2 − 5 0 − 1 3 2 0 1 − 1 0 0 , 所以 A A A R ( A ) = 3 R( A )=3 R ( A ) = 3
矩阵秩的求法:拿到矩阵后，做初等行变换，化为阶梯形矩阵，最后数一下阶梯是几行，他的秩就是几。
矩阵秩的性质总结 1．设 A A A m × n m \times n m × n A A A r r r r r r r r r r r r A A A C m r C n r C_m^r C_n^r C m r C n r r r r A A A r r r r + 1 r+1 r + 1 r r r A A A r ( A ) = r r(A)=r r ( A ) = r A → m × n A \rightarrow m \times n A → m × n r ( A ) ⩽ m . r ( A ) ⩽ n \quad r(A) \leqslant m . \quad r(A) \leqslant n r ( A ) ⩽ m . r ( A ) ⩽ n r ( A ) ⩽ min { m , n } r(A) \leqslant \min \{m, n\} r ( A ) ⩽ min { m , n }
①满阶矩阵。设 A A A n n n ∣ A ∣ ≠ 0. r ( A ) = n |A| \neq 0 . \quad r(A)=n ∣ A ∣ = 0. r ( A ) = n ( A ) = 0. r ( A ) < n (A)=0 . \quad r(A)<n ( A ) = 0. r ( A ) < n α = ( a 1 a 2 ⋮ a n ) \alpha=\left(\begin{array}{c}a_1 \\ a_2 \\ \vdots \\ a_n\end{array}\right) α = a 1 a 2 ⋮ a n r ( α ) ⩽ 1 r(\alpha) \leqslant 1 r ( α ) ⩽ 1
若 α = 0. r ( α ) = 0 \alpha=0 . \quad r(\alpha)=0 α = 0. r ( α ) = 0 α ≠ 0. r ( α ) = 1 \alpha \neq 0 . \quad r(\alpha)=1 α = 0. r ( α ) = 1
A = ( 1 1 1 − 1 3 1 2 1 1 2 3 0 4 ) → r 3 − 2 r 1 r 2 − r 1 ( 1 1 − 1 3 0 1 2 − 2 0 1 2 − 2 ) = ( 1 1 − 1 3 0 1 2 − 2 0 0 0 0 ) A=\left(\begin{array}{ccccc}
1 & 1 & 1 & -1 & 3 \\
1 & 2 & 1 & 1 \\
2 & 3 & 0 & 4
\end{array}\right) \xrightarrow[r_3-2 r_1]{\substack{r_2-r_1}}\left(\begin{array}{cccc}
1 & 1 & -1 & 3 \\
0 & 1 & 2 & -2 \\
0 & 1 & 2 & -2
\end{array}\right)=\left(\begin{array}{cccc}
1 & 1 & -1 & 3 \\
0 & 1 & 2 & -2 \\
0 & 0 & 0 & 0
\end{array}\right) A = 1 1 2 1 2 3 1 1 0 − 1 1 4 3 r 2 − r 1 r 3 − 2 r 1 1 0 0 1 1 1 − 1 2 2 3 − 2 − 2 = 1 0 0 1 1 0 − 1 2 0 3 − 2 0 注：①矩阵秩的本质为方程组约束条件的个数
②r ( A ) = 0 ⟺ A = 0 r(A)=0 \Longleftrightarrow A=0 r ( A ) = 0 ⟺ A = 0 r ( A ) ⩾ 1 ⇔ A ≠ 0 r(A) \geqslant 1 \Leftrightarrow A \neq 0 r ( A ) ⩾ 1 ⇔ A = 0 r ( A ) ⩾ 2 ⇔ A r(A) \geqslant 2 \Leftrightarrow A r ( A ) ⩾ 2 ⇔ A
性质：
（1）r ( A ) = r ( A ⊤ ) = r ( A ⊤ A ) = r ( A A ⊤ ) r(A)=r\left(A^{\top}\right)=r\left(A^{\top} A\right)=r\left(A A^{\top}\right) r ( A ) = r ( A ⊤ ) = r ( A ⊤ A ) = r ( A A ⊤ )
（2）设 A A A B B B γ ( A ± B ) ≤ γ ( A ) + γ ( B ) \gamma(A \pm B) \leq \gamma(A)+\gamma(B) γ ( A ± B ) ≤ γ ( A ) + γ ( B ) A + B . A − B A+B . A-B A + B . A − B r ( A ) + r ( B ) r(A)+r(B) r ( A ) + r ( B )
（3）设 A A A m × n m \times n m × n B B B n × s n \times s n × s r ( A B ) ≤ min { r ( A ) ⋅ r ( B ) } r(A B) \leq \min \{r(A) \cdot r(B)\} r ( A B ) ≤ min { r ( A ) ⋅ r ( B )} { r ( A B ) ⩽ r ( A ) r ( A B ) ⩽ r ( B ) \left\{\begin{array}{l}r(A B) \leqslant r(A) \\ r(A B) \leqslant r(B)\end{array}\right. { r ( A B ) ⩽ r ( A ) r ( A B ) ⩽ r ( B )
（4）设 A A A m × n m \times n m × n B B B n × s n \times s n × s A B = 0 A B=0 A B = 0
r ( A ) + r ( B ) ⩽ n A B = 0 \begin{gathered}
r(A)+r(B) \leqslant n \\
A B=0
\end{gathered} r ( A ) + r ( B ) ⩽ n A B = 0 （5）设 A A A m × n m \times n m × n P ⋅ Q P \cdot Q P ⋅ Q m m m n n n
r ( A ) = r ( P A ) = r ( A Q ) = r ( P A Q ) r(A)=r(P A)=r(A Q)=r(P A Q) r ( A ) = r ( P A ) = r ( A Q ) = r ( P A Q ) （6）没 A A A n n n r ( A ∗ ) = { n , r ( A ) = n 1 , r ( A ) = n − 1 ( n ⩾ 2 ) 0 , r ( A ) < n − 1 r\left(A^*\right)= \begin{cases}n, & r(A)=n \\ 1, & r(A)=n-1 \quad(n \geqslant 2) \\ 0, & r(A)<n-1\end{cases} r ( A ∗ ) = ⎩ ⎨ ⎧ n , 1 , 0 , r ( A ) = n r ( A ) = n − 1 ( n ⩾ 2 ) r ( A ) < n − 1
（7）①设 A , B A, B A , B m × n , n × s m \times n, n \times s m × n , n × s
max { r ( A ) ⋅ r ( B ) } ≤ r ( A B ) ≤ r ( A ) + r ( B ) \max \{r(A) \cdot r(B)\} \leq r\binom{A}{B} \leq r(A)+r(B) max { r ( A ) ⋅ r ( B )} ≤ r ( B A ) ≤ r ( A ) + r ( B ) 或设 A A A B B B m × n . m × s m \times n . m \times s m × n . m × s
max { r ( A ) ⋅ r ( B ) } ⩽ r ( A ⋮ B ) ⩽ r ( A ) + r ( B ) \max \{r (A) \cdot r(B)\} \leqslant r(A \vdots B) \leqslant r(A)+r(B) max { r ( A ) ⋅ r ( B )} ⩽ r ( A ⋮ B ) ⩽ r ( A ) + r ( B ) ②
r ( A 0 0 B ) = r ( A ) + r ( B ) r\left(\begin{array}{ll}
A & 0 \\
0 & B
\end{array}\right)=r(A)+r(B) r ( A 0 0 B ) = r ( A ) + r ( B ) （8）设 A A A n n n r ( A ) = 1 r(A)=1 r ( A ) = 1 α \alpha α β \beta β A = α β T A=\alpha \beta T A = α βT